novel fractal antenna arrays for satellite … · application of the fractal technique to the...

$: NOVEL FRACTAL ANTENNA ARRAYS FOR SATELLITE … · application of the fractal technique to the design of an antenna array can produce radiating systems of large size, and as a consequence$
Progress In Electromagnetics Research, PIER 103, 115–138, 2010

NOVEL FRACTAL ANTENNA ARRAYS FOR SATELLITENETWORKS: CIRCULAR RING SIERPINSKI CARPETARRAYS OPTIMIZED BY GENETIC ALGORITHMS

K. Siakavara

Radiocommunications LaboratoryDepartment of PhysicsAristotle University of Thessaloniki54124 Thessaloniki, Greece

Abstract—A novel fractal antenna-array type is proposed. The designis based on the Sierpinski rectangular carpet concept. However, thegenerator is a circular ring area, filled with radiating elements, sothe higher stages of the fractal development produce large arrays ofcircular rings which, besides the high directivity, have the advantageof the almost uniform azimuthal radiation pattern, attribute thatmany applications require. The introduced arrays can operate asdirect radiating multi-beam phased arrays and meet the requirementsof satellite communications links: high End of Coverage (EOC)directivity, low Side Lobe Level (SLL) and high Career to Interferenceratio (C/I). These operational indices were further optimized by asynthesized multi-objective and multi-dimensional Genetic Algorithm(GA) which, additionally, gave arrays no more than 120 elements.

1. INTRODUCTION

The inspired combination of fractal geometry with the electromagnetictheory has led to the development of an innovative class of antennas,the fractal antennas. The fractal antenna engineering research isfocused on two basic areas: the first deals with the analysis and designof fractal radiating elements, and the second concern the applicationof the fractal theory and the design of antenna arrays [1–3]. Bothantenna types have attributes highly desirable in military as well asin commercial sectors. The fractal antenna elements, the majority ofthem being printed configurations, have compact size, low profile andcost, multi-band operation, easy feeding and, potentially, optimized

Corresponding author: K. Siakavara ([email protected]).

116 Siakavara

operation by suitable modification [4–9]. On the other hand, theapplication of the fractal technique to the design of an antenna arraycan produce radiating systems of large size, and as a consequence ofhigh gain, frequency-independent or multi-band characteristics andradiation patterns with low side-lobe level [1–3, 10]. Moreover, theycan be fed one by one, thus operating as Direct Radiating Antennas(DRAs) and can function as phased arrays without the disadvantages ofthe classical systems of Focal Array Fed Reflectors (FAFRs) used in thepast. Due to these benefits a fractal array could be candidate to serve amodern satellite communication network, for these systems need largeantennas, which illuminate the area under coverage with overlappedbeam spots with high Maximum and End of Coverage Directivity(DEOC), high Carrier to Interference (C/I) ratio and low Side LobeLevel (SLL). For all that, the required high directivity can be obtainedby antennas with large size, and this fact involves large numberof elements and potentially large inter-element distances. So, thedesigners face two problems a) the grating lobes that inevitably comefrom the large distance between the elements, and b) the complexityas well as the high cost of the system due to the large number of theelements that necessitate large number of phase shifters and variousmodules.

Methods to confront the above problems have been proposed. Awidely used technique is to separate the array in sub-arrays, fed byproperly weighted excitations [11, 12]. Innovative techniques, basedon the architecture of the arrays, that effectively solve both theproblems of grating lobes and the large number of control pointshave been recently proposed: a) the sparse arrays synthesized byuniformly excited elements located onto a non-regular grid and b) thethinned arrays that are produced from initial regular arrays by properlywithdrawing a certain number of elements. All the modificationsimposed on the arrays are done via stochastic or deterministicprocedures [13–22]. Moreover, combination of the above concepts haveled to evolutionary methods, which by arranging the array elements inthin-aperiodic lattices have dealt with the problems successfully [23–27].

The fractal technique has been proposed as a separate alternativetechnique for the design of a DRA configuration that meets therequirements of low side lobes and small number of control points.The fractal technique competes with the other methods because itpermits to realize arrays with very small number of elements, as can becombined with a) the thinning procedure [28] b) the genetic algorithms[29] or c) the method of clustering the array in sub-arrays with non-uniform excitation [30].

Progress In Electromagnetics Research, PIER 103, 2010 117

In conclusion, the innovative design techniques, deterministic ornot, give solutions to one of the most serious problems of satelliteantenna arrays, namely to control the SLL of their radiation patterns.In some cases the design methods, followed by an optimizationtechnique, can lead also to an efficient control of the Carrier toInterference ratio. Almost all the proposed configurations suffer fromthe disadvantage of the large number of elements and the complexityof the feeding network. The target of the present work was tosynthesize satellite direct radiating array configurations which besidesthe attributes of the low side lobes and the large C/I, would haveelements, the number of which will be not greater than some tensand that would need a simple feeding network. The synthesis of theproposed arrays is based on the rectangular Sierpinski carpet concept.However, it is realized by a novel configuration made up of circularrings. The first and second stages of growth are studied. Both stageslead to operational characteristics that meet the requirements of asatellite coverage. The drawback of the array of the second stage isthe large number of elements that makes it non-competitive to othertypes of large antennas. On the contrary, the first stage produces anarray with small number of elements and a geometry that is susceptibleto improvement, in order the features of the radiation pattern to befurther optimized. For the optimization, a genetic algorithm processwas chosen. Due to the nature of the problem, namely the numberof the objectives that had to be optimized and the number of theparameters, a specific genetic algorithm (GA) was synthesized andcould be classified in the category of Multi-Objective and Multi-Dimensional GAs.

2. THE CIRCULAR RING SIERPINSKI-CARPETFRACTAL ARRAY

The fractal technique is based on the idea of realizing the radiationcharacteristics by repeating a structure in arbitrary or regular scales.The basic scheme of a fractaly designed radiating system is a generatingsub-array. In particular, the entire array can be formed recursivelythrough repetitive application of the generating sub-array under aspecified scaling factor which is one of the parameters of the problem.This process is realized following potentially two different strategies:By one of them, the repetition of the generating sub-array is madein such a way that the size and number of the elements of the arrayget larger from stage to stage. So, at the nth stage, the size of thearray is equal to the size of the generating sub-array multiplied by thenth power of the scaling factor, and the elements’ population is the

118 Siakavara

(n + 1)th power of the number of elements of generating array. Bythe second strategy, the entire area, which the final array is permittedto occupy, is defined a priori. Then, by the process of the properrepetition of the generator, the available area is filled by scaled replicasof the generator. In this case solely the number of the elements of thearray becomes larger from stage to stage, whereas the total size of thearray does not change. The Sierpinski fractal arrays belong to thistype of fractal configurations.

The Sierpinski rectangular carpet and triangular gasket have beenefficiently used in the past for the realization of antenna arrays withspecific operational features [1, 2, 31]. Both of these types, althoughcan produce arrays with high directivity and potentially low side-lobes, cannot give radiation patterns with azimuthal uniformity. In thepresent work a novel approach is introduced: following the Sierpinskiconcept, however with another shape of generator. The geometry,which would fulfill the requirement of azimuthally uniform radiationpattern, is the circular ring, and this shape was used as generator.

2.1. The Array Synthesis

The novel array, in accordance to the rectangular carpet concept,is developed as follows: To define the correspondence betweenthe rectangular carpet and the proposed array, suppose that thepermissible area for the rectangle array has size Lt × Lt. Theconfiguration of the generator is shown in Fig. 1(a). The shadowedarea represents the region in which the radiating antenna elementscould be positioned whatever shape or size would have. The remainingrectangular white sub-areas are empty of elements or have elementsthat are turned off. Equivalently, the entirely available area for thenovel array would be a circular disc of radius Rt (Fig. 1(b)). Theshadowed area, in this configuration instead of a rectangle, would be acircular ring of width wo. If we consider that the internal radius is ro,the width wo is defined as wo = Rt−ro

3 . This definition is in agreementwith the rectangular carpet in which LS = Lt

3 . The remaining area isdivided in rings of width equal to wo as shown in Fig. 1(b). These ringsare empty or have antennas that are turned off whereas, the centralring is full of radiating elements.

Figures 2(a) and 2(b) depict the first stage of fractal developmentof the rectangular and circular ring carpet. In the circular ring case,each one of the rings is divided in three sub-rings of equal width,w1 = wo

3 . So, a total of three blocks, each one with three rings,is formed, and the number of all the rings is Nr2 = 32. All threesub-rings of the block around the center of the array are filled with


(a) (b)

Figure 1. The generator subarray (a) rectangular, (b) circular ring,filled with antenna elements.

antenna elements which are turned on. These elements would becircular apertures or have another shape. In the present work thecircular aperture was chosen. In accordance to the Sierpinski conceptthe central sub-rings of the other two blocks are filled with radiatingelements whereas the remaining sub-rings have turned-off elements orare empty of antennas.

Following the same procedure, the second stage of fractaldevelopment is produced (Fig. 3(c)). In the circular ring fractal array,each of the sub-rings of the previous stage is also divided into threerings of equal width, which is w2 = w1

3 = wo9 . In this way, the array

is formed by a total of nine blocks, each one having three rings ofwidth w2 and the total number of rings being Nr3 = 33. The centralthree of Nr3, the 5th and 8th block areas are full of turned on antennaelements. In the 4th, 6th, 7th and 9th blocks, the central sub-ring isfilled with elements that are turned on with size smaller than that ofthe other elements. The remaining area of the array is empty. So, theconfiguration of the second stage of growth has five rings with largeand four rings with small elements.

In all antennas a unique element would be placed in the empty areaaround the center of the arrays. This placement is somehow arbitrary,is by the decision of the designer and would enhance the performanceof the arrays a little.

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(a)

(b) (c)

Figure 2. The geometry of the 1st stage of growth. (a) Therectangular configuration, (b) the circular configuration.

2.2. The Fields

The array factor and the total field of the first stage of growth(Fig. 2(c)), in accordance with the antenna array theory [32], aredescribed by Equations (1) and (2) respectively,

AF1(θ, ϕ) = Io +5∑

i=1

Ni∑

j=1

IijejkRi sin θ cos(ϕ−ϕij) (1)

E1tot(θ, ϕ) = AF1(θ, ϕ) · Eel(θ, ϕ) (2)

where Ri is the radius, and Ni is the number of elements of the ithring. ϕij and Iij are, respectively, the location and excitation of the jthelement on the ith ring, and the coefficient Io is the excitation of the


(a)

(b) (c)

Figure 3. The geometry of the 2nd stage of growth. (a) Therectangular configuration, (b) the circular configuration, (c) thecircular ring areas filled with antenna elements.

element located at the center. Eel(θ, ϕ) is the field of each radiatingelement. If the central element has size different from the rest of theelements, the coefficient Io is multiplied separately by the respectiveelement field.

The total field for the second stage configuration (Fig. 3(c)) iscalculated by Equation (3)

E2tot(θ, ϕ) = Io ·E◦el(θ, ϕ)+

5∑

i=1

Nì∑

j=1

Iìje

jkRì sin θ cos

(ϕ−ϕ`

ij

)·E`

el(θ, ϕ)

+

4∑

i=1

Nsi∑

j=1

Isije

jkRsi sin θ cos(ϕ−ϕs

ij

) · Es

el(θ, ϕ) (3)

In the above equation, the terms of the total summation aredivided in three groups depending on the size of the antenna elements:

122 Siakavara

the central element with field E◦el(θ, ϕ), the group of the elements with

large size and field E`el(θ, ϕ) and the elements with small size and field

Esel(θ, ϕ).

Both the above array schemes would serve, as mentionedpreviously, a satellite system. The common practice in theseapplications is to synthesize an antenna array having the propersize, configuration and excitation to produce narrow main beams ofradiation that create on the earth spots of illumination which areoverlapped and with very small angular width. Moreover, inside theterrestrial area under coverage there are spots that use the samefrequency band, and the angular distance between them is very small.So, the possibility of interference is very high. The solution to thisproblem is given by the antenna array via suitable modification of itsradiation pattern in order the radiation toward the co-channel spotsto be minimized. Furthermore, another general requirement concernsthe level of the radiation lobes appearing in the range out of coveragearea — on earth as well as out of earth. This level must be as smallas possible.

As an application example of the proposed design method, anarray was synthesized with the requirement to produce on the earthspots with 0.65◦ angular width and to cover, via overlapped spots, analmost circular area with angular width 2 · 1.445◦. The first step ofthe design is to select the radius Rt of the entire array. The larger itis, the larger would be the directivity, but there is a tradeoff betweenthis quantity and the size available for satellite antennas. Furthermore,a very large antenna will have undesirably large number of elements.If we approximately consider the entire array as a circular uniformlyexcited aperture, the radius Rt, in wavelengths, and the half-powerbeam-width HP ◦ are related with the equation Rt ' 29/HP ◦ [29].For HP ◦ = 0.65◦, Rt ' 45λ. However, the circular ring fractalarrays have no uniform excitation because there are empty areas andelements with different sizes. So, a larger radius is necessary. Somecalculations of the radiated field proved that in order the directivity toexceed the value of 40 dB a radius about 68λ was necessary. Followingthe procedure described above, the arrays of Figs. 2(c) and 3(c) wereproduced. The arithmetic values of their geometrical parameters areincluded in Table 1. In both cases the large elements of the arraywere suggested to be circular apertures of uniform excitation with radiir`el = 3.4λ, whereas the small elements are uniformly excited apertures

with radii rsel = 1.2λ. At the center of both arrays a unique circular

aperture, uniformly excited with radius r◦el = 4.4λ, was located.The directivity patterns of the arrays are computed by


Table 1. The geometry of the 1st stage (Fig. 2(c)) and the 2nd stage(Fig. 3(c)).

1st stage 2nd stage 1st stage 2nd stageR1 8.4λ R`

1 8.4λ N1 7 N `1 7

R2 15.5λ R`2 15.5λ N2 13 N `

2 13R3 22.5λ R`

3 22.5λ N3 20 N `3 20

R4 35.9λ R`4 35.9λ N4 32 N `

4 32R5 56.6λ R`

5 56.6λ N5 51 N `5 51

Rs1 29λ N6 N s

1 72Rs

2 42.7λ N7 N s2 104

Rs3 49.5λ N8 N s

3 114Rs

4 64.3λ N9 N s4 150

Ntotal 123 + 1 563 + 1

Equation (4a)

D (θ, ϕ) = Dmax |Etot(θ, ϕ)|norm (4a)

where

Dmax =4π

2π∫0

π∫0

(|Etot(θ, ϕ)|norm)2 sin θdθdϕ

(4b)

and

|Etot(θ, ϕ)|norm = |Etot(θ, ϕ)|/|Etot(θmax, ϕmax)| (4c)

The central beams of the above arrays and respective spots are atθ = 0◦. The radiation patterns of these beams are depicted in Figs. 4and 5. The calculations were made via Equations (2) to (4), withMathCad software. In order to check the azimuthal uniformity of thepatterns, ϕ-cuts from 0◦ to 90◦ per one degree were calculated. All theelements in both arrays were fed by currents of equal value, and therecords of the arrays are presented in Table 2.

The End of Coverage Directivity (DEOC [dBi]) is the directivityat θ = 0.65◦/2. For every spot a co-channel spot is located 2 ∗ 0.56◦apart, and it is required the radiation pattern to have a deep minimumin the range 0.795◦ ≤ θ ≤ 1.445◦. This minimum is assessed by thequantity C/I, or more simple CI, defined as the ratio DEOC/(max

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Table 2. The records of the initial circular ring Sierpinski carpets.

Dmax [dBi] DEOC [dBi] CI [dB] SLL [dB]1st fractal stage 45 39.95 16.1 18.82nd fractal stage 44.95 37.18 7.18 21

level inside co-channel region). The range ‘out of coverage-on earth’corresponds to angles θ between 1.445◦ and almost 8◦. In this rangeas well as for θ > 8◦ the ratio of the peak level of directivity over theside lobe is termed as SLL. In most of the applications it has to begreater than 20 dB. The quantity CI [dB] and SLL [dB] are measuredas shown in Fig. 4.

From the results, it is obvious that the array of the second stageis inferior to that of the first stage. It has much more elements andsmaller DEOC , CI and SLL. However, in practice, large values forall the indices, DEOC , CI, SLL, are required. The performance ofboth arrays could be improved applying an optimization technique.The challenge is to try to optimize the array of the first stage

Figure 4. Patterns of the central beam of the array of the first stageof growth (Fig. 2(c)), ϕ-cuts from 0◦–90◦. All the elements are excitedwith equal currents.


Figure 5. Patterns of the central beam of the array of the secondstage of growth (Fig. 3(c)), ϕ-cuts from 0◦–90◦. All the elements areuniformly excited.

due to its small number of elements, and the target is, along withthe improvement of the indices of operation, to further reduce thenumber of the elements. For the optimization, the GA method wasselected, for reasons discussed in the following. The parameters thatgovern the performance of the array are the geometry and excitation.So, the target of the optimization procedure was to find a newconfiguration, coming from the initial one, as well as a suitable non-uniform excitation.

3. OPTIMIZATION BY GENETIC ALGORITHMS

Genetic Algorithms have been widely used in antenna synthesis andoptimization problems [33, 34]. They have been proven efficient tofinding the proper excitations or the suitable division of the arrays insub-arrays in order the antenna to produce a desired radiation patternwith respect to the SLL and the directions of the nulls or to optimizethe antenna by broadening the frequency bandwidth [11, 15–17, 33–35].GAs have also been used to thin an array [18], in combination withParticle Swarm Optimization Technique to modify sparse arrays [21]or to modify fractal arrays [29, 36]. In all cases the target was theachievement of special performance features.

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In the work at hand, a GA was produced to optimize theradiation features of the proposed circular ring Sierpinski carpetantenna array, in order to exhibit the features which are necessaryfor a satellite communication radiating system. The problem wascomplex, because the target was to modify the array in order to obtainthe contemporaneous optimization of three quantities, the DEOC , C/Iand SLL. So, the problem was Multi-Objective. On the other side,the aim of the modification of the array was to find an optimumgeometry and an optimum excitation. So, several parameters had tobe optimized. This means that the optimization had to be multi-dimensional. These are the reasons for which a GA procedure wasselected for the optimization: the GAs can deal with a large numberof parameters and objectives. Moreover, in the problem at hand, thereis a trade-off among the objectives. For example, the values of theparameters that ensure high DEOC fail to give large C/I and SLL, andvice versa, or, satisfactory values could be obtained for DEOC and SLLbut not for the C/I. As a consequence, the problem would not have aunique solution, and a set of equally valid and alternative solutions isexpected. It is an additional reason for which the GA procedure wasselected for the specific problem: a GA has the potential of looking formore than one solutions in parallel.

3.1. Formulation

A general multi-objective and simultaneously multi-dimensional designproblem can be expressed as the maximization of an objective vector

g(p̄) = [g1(p̄), g2(p̄), . . . , gM (p̄)] (5)

where p̄ = [p1, p2, . . . , pN ] is the vector, also termed chromosome,containing the parameters of the problem, and gi(p̄), i = 1, . . . , M arethe objective functions termed cost functions. The objective vectorrepresents the mapping between the search space and the space of theobjectives. In the present problem the cost functions are three: theDEOC(p̄), CI(p̄) and SLL(p̄), and they are considered in the entirespace out the coverage area. The radii R3, R4, R5, (Fig. 2(c)), thenumbers N3, N4, N5 of the elements of the respective rings and theexcitation were selected as the parameters which would maximize thecost functions. The strategy was to keep the radii R1, R2 unchangedbecause it is expected that high density of elements at the centralarea of the array would help the suppression of side lobes. It isalso known that this attribute would be enhanced by non uniformexcitation, namely by feeding the central elements of the array withcurrents larger than the currents of the rest elements. In the arrayunder optimization, the elements of the rings with radii R1, R2, R3,


R4 were fed with current values Po times the currents of the outer ring.The parameter Po is the 7th parameter of the chromosome p̄, whichhas the form

p̄ = [R3, R4, R5, N3, N4, N5, Po] (6)

The multi-objective genetic procedure, in its classical approach,requires the maximization or minimization of the square mean valueof the vector g(p̄). In our case the maximization of this quantity doesnot ensure that all the objectives DEOC(p̄), CI(p̄) and SLL(p̄) willhave contemporaneous satisfactory high values. So, in the synthesizedalgorithm these objectives worked as three separate cost functions. Theideal solution p̄ would be the one that could simultaneously maximizeall three cost functions, but due to the trade-off among them it seemsrather impossible. Each cost function has its own cost surface as wellas its own global maximum, but, possibly, these maxima would notcorrespond to the same chromosome p̄. Moreover, the global maximumvalues of the objectives DEOC , CI and SLL of the proposed arrayconfiguration are not known a priori.

So, the GA had to be synthesized in a way that would makeit capable to explore the trade-offs of the objectives and to work asan unsupervised procedure, capable to converge not necessarily to aunique solution but to a set of optimal solutions of equal valid.

The code of the synthesized GA was realized with FORTRAN,keeps the attributes of a classical multi-objective algorithm, and isenhanced with elitist strategies and criterions that work as thresholds.All of them decide on how many and which of the individuals-chromosomes will pass from one generation to the next. The numberof the chromosomes at the ith iteration-generation is termed as N i

pop.The GA includes the following steps:Step 1: Generate an initial population of N1

pop chromosomes p̄.Step 2: Calculate the cost functions for all chromosomes,

DEOC(p̄i), CI(p̄i) and SLL(p̄i), i = 1, 2, . . . , N1pop

Step 3: Select N imat ≤ N i

pop individuals for mating. Thisnumber is one of the parameters of the algorithm. It is calculatedas a fraction mf of the population of the ith iteration, and it is thesame for all the iterations. So, N i

mat = mf · N ipop, where N i

pop is thenumber of individuals in the ith generation. The selection of which ofthe chromosomes amongst the individuals of the ith population willconstitute pairs to produce two offspring is made randomly.

Step 4: Execute the process of mating. The classical approachis to randomly select crossover points in the chromosomes of theparents and exchange the parameters. In the problem at handthis procedure could not introduce new information because eachcontinuous parameter, which was randomly initialized in the first

128 Siakavara

population, is transferred to the next generation simply in differentcombinations. So, another technique was applied. The nth parameter,pi

n,off , of the offspring, at the ith generation is produced by thelinear combination of the respective parameters of its parents. Theparameters of the first offspring are produced via the rule

pin1,off , = βpi

n,mom + (1− β)pin,dad (7)

and of the second offspring by

pin2,off = (1− β)pi

n,mom + βpin,dad (8)

where β is a random number in the interval [0, 1], and pin,mom, pi

n,dad

are the nth parameters in the mother’s and father’s chromosomes.Step 5: Calculate the three cost functions for each offspring,

namely DEOC(p̄ioff ), CI(p̄i

off ) and SLL(p̄ioff ).

Step 6: Impose the rules to select which of the individuals —chromosomes will survive and constitute the next generation. Thetarget of this selection is to discard the chromosomes that potentially,during the mating, would produce offspring with degrade.

a. At first, an elitism criterion is imposed. The values of all thethree cost functions produced by the offspring are compared with therespective of their parents. The offspring replace their parents andpass to the (i + 1)th generation by the rule,

p̄i+1n =

p̄in,off if

DEOC(p̄in,off ) ≥ DEOC(p̄i

n,par) ∧ CI(p̄in,off )

≥ CI(p̄in,par) ∧ SLL(p̄i

n,off ) ≥ SLL(p̄in,par)

p̄in,par otherwise

(9)

where DEOC(p̄in,par), CI(p̄i

n,par) and SLL(p̄in,par) are the values of the

cost functions of the respective parents.b. A threshold Di

EOCthr, CIithr and SLLi

thr, for every cost value,is established, and if we define the populations of the ith and (i+1)thiterations as P i

pop and P i+1pop , then

∀n ∈ (1, 2, . . . , N ipop), p̄i

n ∈(P i

pop ∩ P i+1pop

)

if DEOC(p̄in)≥Di

EOCthr∧CI(p̄in)≥CIi

thr∧SLL(p̄in)≥SLLi

thr (10)

The thresholds are not constant. They increase, by a step, from oneiteration to the next, starting from values selected to be equal to thoseof the initial array of the first stage of growth (Fig. 2(c)). In this way,the number of chromosomes in the populations gradually decreases,and better performers are included from generation to generation.

Step 7: As mentioned previously, the global maxima of each costfunction are not known a priori and potentially could not be obtainedby the same chromosome. So, the GA can stop when a final population


of Nstop individuals is found, provided that these individuals ensurelarge values for all three cost functions. The genetic procedure stopswhen N i

pop ≤ Nstop, else the process goes back to Step 3. The flowchartof the synthesized GA is shown in Fig. 6.

3.2. Results

The genetic procedure described above was applied to an initialpopulation with N1

pop = 720 chromosomes p̄, produced from the fractalarray of the first stage of growth (Fig. 2(c)). This relatively largenumber was chosen in order the algorithm to sample the cost surfacesin more detail. The individuals of the initial population were createdrandomly, with respect to the parameters of the chromosomes underthe single constraint, the number of the elements in the rings to beless or equal to the maximum permissible. By this constraint it wasensured that the elements do not overlap, whereas, by the randomselection, chromosomes with small number of elements were included inthe population, and in this way the potential to find the GA solutionswith number of elements smaller than that of the initial array wasprovided.

Other parameters that affect the performance of the GA werethe steps by which the thresholds were increased. The target of thisincrease, as mentioned before, was to reject, from one generation to thenext, the most ‘weak’ chromosomes, in the sense of the capability toensure large values of the cost functions. This evolution procedure hadto be made by a proper rate. If it was too fast, the populations woulddecrease quickly, and this fact would reduce the capability of the GAto converge to solutions that provide satisfactory high values of costfunctions. On the other hand, a slow process would potentially leadto bad performers, the chance to contribute their traits to the nextgeneration, degrading the capability of the algorithm to find goodsolutions. After many trials it was found that the best results areobtained with steps 0.05 dBi for the DEOC, 0.15 dB for the CI and0.1 dB for the SLL.

To keep track of the evolution of the generations, the number ofthe chromosomes and mean values Di

EOC/mean, CIi

mean and SLLimean

calculated on all chromosomes of the ith generation were found andstored. The results are depicted in Figs. 7(a) to 7(d), as a function ofthe order of generation and for various values of the parameter mf .

In Figs. 7(a)–7(d), it is shown that each of DiEOC/mean

, CIimean

and SLLimean tends to a maximum value but not for the same value of

mf . This fact is a result of the tradeoff among the cost functions. Ithas to be pointed out that the mean values were a criterion to judge if

130 Siakavara

Compare offspring with

parents and replace or not

Increase the thresholds and

impose the criterions to all the

individuals of the population

New generation

Check the population with the

stop criterion

STOP

Calculate the three cost functions for all offspring

Select randomly a fraction of

the population for mating

Execute mating by the linear rule with random coefficients

Create the initial population

Evaluate cost functions for all chromosomes

Figure 6. The GA process.


0

50

100

150

200

250

300

350

400

0 5 10 15 20 25 30 35 40 45 50

20

22

24

26

28

30

32

CI

[dB

]

5 10 15 20 25 30 35 40 45 5 0

(c) (d)

po

pu

latio

n

generationgeneration

41.5

41.6

41.7

41.8

41.9

42.0

42.1

42.2

42.3

42.4

42.5

DE

OC [dB

i]

15

16

17

18

19

SLL

[dB

]

0 5 10 15 20 25 30 35 40 45 50

(a)

0 5 10 15 20 25 30 35 40 45 50

(b)generation generation

mf=1

1/2

1/3

1/4

1/5

mf=1

1/2

1/3

1/4

1/5

mf=1

1/2

1/3

1/4

1/5

mf=1

1/2

1/3

1/4

1/5

0

Figure 7. Results as function of the order of generation. (a) Numberof individuals, (b) mean values of the End of Coverage Directivity, (c)mean values of the Carrier to Interference ratio, and (d) mean valuesof the SLL.

the generations tend to a better performance. So, independent of themean values that would be larger or smaller, depending on the valueof mf , in the final population, and for every mf , chromosomes, whichensure large values for all three cost functions, appear. Representativeresults, considered as the best, are included in Table 3.

Comparing the results of Tables 2 and 3, the general ascertainmentis that the GA obtained to give solutions with total number of elementssmaller than that of the initial array, DEOC greater, at least 2 dB, CIgreater, at least 11 dB, and SLL greater, at least 3 dB, than those ofthe initial array. It is observed that the solutions with a little largerDEOC have smaller CI and SLL and vice-versa. Furthermore, for thesmaller values of mf the GA converges to solutions with larger numberof elements, a little smaller values of SLL and significantly smallervalues of CI. The size of all the arrays is ∼ 2R5, almost similar inall solutions. The large values of mf , namely the strategy to mate the

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most or all the individuals of each generation, allows the algorithm toconverge with smaller number of iterations, as shown in Fig. 6(a), andalso to search for and find solutions with better characteristics.

The first two solutions of Table 3 could be considered as the best of

(a)

(b)

Figure 8. Solution 1: (a) The layout, outer radius 54.91λ, 107elements. (b) Patterns of the central beam ϕ-cuts from 0◦–90◦.DEOC = 42.04 dBi, CI = 32.07 dB, SLL = 21.78 dB.


(a)

(b)

Figure 9. Solution 2: (a) The layout, outer radius 54.423λ, 113elements. (b) Patterns of the central beam ϕ-cuts from 0◦–90◦.DEOC = 42.21 dBi, CI = 32.25 dB, SLL = 22.59 dB.

all. The details of their geometry are presented in Table 4. The layoutsand radiation patterns are depicted in Figs. 8 and 9. The calculationswere made by applying the parameter values resulted from the GA,Equations (2)–(4) and MathCad software.

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Table 3. The best solutions resulted from the GA.

solution mf DEOC [dBi] CI [dB] SLL [dB]

Total

number of

elements

R5/λ Po

1 1 42.04 32.07 21.78 107 54.91 2

2 1 42.21 32.25 22.59 113 54.42 2

3 1/2 42.27 31.94 21.83 118 55.05 2

4 1/3 42.3 29 21.82 120 54.65 2

5 1/4 42.56 27.32 21.91 128 55.00 2

6 1/5 42.56 27.32 21.92 128 55.00 2

Table 4. The geometry of the first two solutions of Table 3.

Solution 2 Solution 1 Solution 2 Solution 1R1 8.4λ 8.4λ N1 7 7R2 15.5λ 15.5λ N2 13 13R3 26.535λ 26.475λ N3 21 19R4 37.313λ 37.257λ N4 28 28R5 54.423λ 54.91λ N5 43 39

Ntotal 112 + 1 106 + 1

4. CONCLUSIONS

The paper presents a novel process for the design of fractal largeantenna arrays. The process is based on the rectangular Sierpinskicarpet concept but starts from circular ring generator, and as aconsequence the higher stages of fractal growth are arrays of circularring areas filled with radiating elements. The benefits of the proposedantenna are a) the azimuthal uniformity of the radiation pattern,due to the circular configuration of the array b) the high directivitywhich is obtained with relatively small number of elements at the firststage of fractal development and c) the elements can be fed one byone, thus the system operates as a direct radiating and potentiallyphased array. These features would make the proposed arrays a goodchoice for a satellite communication platform. However, in theseapplications additional operational features are required: a) side lobelevels, very low compared to the peak values of the radiation b) highcareer to interference ratio in the co-channel areas of the networkand c) radiating systems with small number of elements and lowfeeding complexity. What makes the design more difficult is that the


co-channel areas are located in a very small angular distance. Anadditional advantage of the proposed arrays, especially that of the 1ststage of growth, is the capability of improvement and, in this way,meeting the above requirements. A hybrid multi-objective and multi-dimensional Genetic Algorithm was synthesized for the optimization ofthe array of the first stage of growth. The algorithm, due to the natureof the problem, does not converge to a unique solution which would bethe globally best one, but to a number of satisfactory solutions of equalvalid. The genetic procedure is obtained to improve the operationalfeatures of the antennas. It increased a) the end of coverage directivityat least 2 dB b) the carrier to interference ratio at least 11 dB and c)the difference between the peak radiation and the level of side lobesat least 3 dB, and besides these improvements, gave arrays of no morethan 120 elements.

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